document.write( "Question 286517: how to factor 3w^2+7w-20 \n" ); document.write( "
Algebra.Com's Answer #207808 by jim_thompson5910(35256)\"\" \"About 
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression \"3w%5E2%2B7w-20\", we can see that the first coefficient is \"3\", the second coefficient is \"7\", and the last term is \"-20\".



Now multiply the first coefficient \"3\" by the last term \"-20\" to get \"%283%29%28-20%29=-60\".



Now the question is: what two whole numbers multiply to \"-60\" (the previous product) and add to the second coefficient \"7\"?



To find these two numbers, we need to list all of the factors of \"-60\" (the previous product).



Factors of \"-60\":

1,2,3,4,5,6,10,12,15,20,30,60

-1,-2,-3,-4,-5,-6,-10,-12,-15,-20,-30,-60



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to \"-60\".

1*(-60) = -60
2*(-30) = -60
3*(-20) = -60
4*(-15) = -60
5*(-12) = -60
6*(-10) = -60
(-1)*(60) = -60
(-2)*(30) = -60
(-3)*(20) = -60
(-4)*(15) = -60
(-5)*(12) = -60
(-6)*(10) = -60


Now let's add up each pair of factors to see if one pair adds to the middle coefficient \"7\":



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First NumberSecond NumberSum
1-601+(-60)=-59
2-302+(-30)=-28
3-203+(-20)=-17
4-154+(-15)=-11
5-125+(-12)=-7
6-106+(-10)=-4
-160-1+60=59
-230-2+30=28
-320-3+20=17
-415-4+15=11
-512-5+12=7
-610-6+10=4




From the table, we can see that the two numbers \"-5\" and \"12\" add to \"7\" (the middle coefficient).



So the two numbers \"-5\" and \"12\" both multiply to \"-60\" and add to \"7\"



Now replace the middle term \"7w\" with \"-5w%2B12w\". Remember, \"-5\" and \"12\" add to \"7\". So this shows us that \"-5w%2B12w=7w\".



\"3w%5E2%2Bhighlight%28-5w%2B12w%29-20\" Replace the second term \"7w\" with \"-5w%2B12w\".



\"%283w%5E2-5w%29%2B%2812w-20%29\" Group the terms into two pairs.



\"w%283w-5%29%2B%2812w-20%29\" Factor out the GCF \"w\" from the first group.



\"w%283w-5%29%2B4%283w-5%29\" Factor out \"4\" from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.



\"%28w%2B4%29%283w-5%29\" Combine like terms. Or factor out the common term \"3w-5\"



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Answer:



So \"3%2Aw%5E2%2B7%2Aw-20\" factors to \"%28w%2B4%29%283w-5%29\".



In other words, \"3%2Aw%5E2%2B7%2Aw-20=%28w%2B4%29%283w-5%29\".



Note: you can check the answer by expanding \"%28w%2B4%29%283w-5%29\" to get \"3%2Aw%5E2%2B7%2Aw-20\" or by graphing the original expression and the answer (the two graphs should be identical).

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